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Interaction vertices

The Table Vertices contains interaction vertices. The first four fields $A1$, $A2$, $A3$, $A4$ include the names of the interacting particles. These fields must contain particle names in CompHEP  notation. $A4$ may be empty. The last two fields 'Factor' and 'LorentzPart' define a vertex itself. Let $S$ be the action, then a functional derivative of $S$ over fields is represented as

$$\frac{\delta S}{\delta A1_{[m1]}(p1)\, \delta A2_{[m2]}(p2)\, \delta A3_{[m3]}(p3)\, [\delta A4_{[m4]}(p4)]} =$$

$$(2 \pi)^4\delta(p1+p2+p3\, [+p4]) \, [\gamma_0] \, ColorStructure \cdot Factor \cdot LorentzPart\;.$$ (8)

Here $p$ and $m$ denote 4-momenta and Lorentz indices. The brackets [ ] are used to mark the optional parts of expression. Thus, $A4$, $p4$, and $m4$ appear only in the case of four particle vertex. In the case of anti-commuting fields the right-side derivatives are assumed. The Fourier transformation is defined by

$$A(x) = \int \! \exp(-i k x) A(k) \,d^4k\;.$$ (9)

'Factor' must be a rational monomial constructed of the model identifiers, integer numbers and imaginary unity.

'LorentzPart' must be a tensor or Dirac $\gamma$-matrix expression. Coefficients of this expression are polynomials of the model identifiers and scalar products of momenta. The division '/' operator is forbidden in 'LorentzPart'. It must be transferred to the 'Factor' field.

Similar to the Reduce  notation, in order to construct scalar products of momenta, momentum components, and metric tensors we use the dot symbol, for example,

$$\begin{eqnarray*} p1.p2 & is & g_{\mu \nu} p_1^\mu p_2^\nu\;; \\ p1.m2 & is & p_1^{m_2}\;;\\ m1.m2 & is & g_{m_1 m_2}\;. \end{eqnarray*}$$

To implement the Dirac $\gamma$-matrix with index $'m'$ we use a symbol $G(m)$, whereas $G(p)$ denotes $p_{\mu}\gamma^{\mu}$. Anti-commutation relations for $\gamma$ matrices should be written as

$$G(v1)\, G(v2) + G(v2)\, G(v1) = 2 \; v1.v2,$$ (10)

where $v1, v2$ are momenta or indices.

The $\gamma_5$ matrix is denoted by $G5$. It is defined by equation

$$\gamma_5 = i\; \gamma_0\gamma_1\gamma_2\gamma_3$$

The number of fermion fields in one vertex must be two or zero. If you would like to implement a four-fermion interaction, use an auxiliary unphysical field which may be constructed by means of the '*' symbol in the 'Aux' column of the particle table (see Section ).

CompHEP  interprets the anti-particle spinor field as the Hermitian conjugated particle field, rather than the Dirac conjugated one. Also it is assumed that all spinor fields are written in the Majorana basis and the matrix of C-conjugation is equal to $(-\gamma_0)$ ¹. After substitution $C \to -\gamma_0$ all possible vertices could be written as

$$A1(p1) \, \gamma_0 \, G(v1) \, G(v2) \ldots G(vn) \, A2(p2)$$

where $A1$ and $A2$ are some spinor fields, each of them corresponding to particle or anti-particle. This form of Lagrangian is assumed in the vertex expression (8). $\gamma_0$ is substituted by CompHEP  automatically in the case of a spinor particle vertex and is not expected in LorentzPart of (8).

Note that structures like $m1.m2$ and $p1.m2$ are forbidden for the vertex with fermions. In order to implement these structures use the equation (10).

Let us note that by definition (8) the LorentzPart has the corresponding symmetry property in the case when identical particles appear in one vertex. This symmetry is not checked by CompHEP, but its absence will lead to wrong results. The following equation may be used to check the symmetry in the case of fermion vertex:

$$A1(p1) \, \gamma_0 \, G(v1) \, G(v2) \ldots [G5] \ldots G(vn) \, A2(p2) =$$

$$A2(p2)\, \gamma_0 \, (-G(vn)) \ldots [G5] \ldots (-G(v2)) \, (-G(v1)) \, A1(p1) \;.$$ (11)

It may be useful also to check the Lagrangian self-conjugation property. Note that the anticommutation of A1 and A2 is already taken into account in (11).

ColorStructure is substituted by CompHEP  automatically. For a colorless particle vertex it is equal to 1. For $(3 \times \bar{3})$ and for $(8 \times 8)$ vertices the unity tensor is substituted. If CompHEP  meets a vertex with three particles in the adjoint representation $(8 \times 8 \times 8)$, it substitutes

$$-\,i\, f(a1,a2,a3),$$

where $f^{\alpha_1}_{\alpha_2 \alpha_3}$ are the structure constants of SU(3). Color indices $a1, a2, a3$ are taken in the same order as they appear in the particle columns. For the $(3 \times \bar{3} \times 8)$ vertex CompHEP  substitutes

$$\frac{1}{2}\lambda(\bar{i},i,a),$$

where $\lambda(\bar{i},i,a)$ are the Gell-Mann matrices. More complicated color structures are not implemented yet, but it is possible to construct them by means of unphysical particles (Aux='*') or tensor ghosts (Section ). In the case of tensor auxiliary field use the capital 'M' for designation of the second Lorentz index of this field as it is shown in equation (7).

Footnote

1  Above the Majorana spinor was defined as a Hermitian self-conjugated one: $\psi=\psi^+$. In the same time it must be $C$-self-conjugated too: $\psi=C \bar{\psi}^T$. Following these requests we get the phase of $C$ which is different from one chosen in the textbook [Bjorken&Drell]